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  "body": "\n## Definition\n\nLet $F$ be a field and let $V$ be a vector space over $F$.\nLet $T(V) = \\bigoplus_{k \\ge 0} V^{\\otimes k}$ be the tensor algebra of $V$, where $V^{\\otimes 0} = F$ and the product is the [[tensor product|0BF2DE2SYK]].\nLet $I$ be the two-sided ideal of $T(V)$ generated by the elements $v \\otimes v$ for $v \\in V$.\nThe *exterior algebra* of $V$ is the quotient algebra $\\Lambda(V) = T(V) / I$.\nThe product in $\\Lambda(V)$ is written $\\wedge$.\n\n## Remarks\n\nFor all $u, v \\in V$, $v \\wedge v = 0$ and $u \\wedge v = -v \\wedge u$.\nThe image of $V^{\\otimes k}$ is the $k$-th exterior power $\\Lambda^k(V)$; if $\\dim V = n$, then $\\dim \\Lambda^k(V) = \\binom{n}{k}$ and $\\dim \\Lambda(V) = 2^n$.\nIf $(e_1, \\dots, e_n)$ is a basis of $V$, then $\\Lambda^n(V)$ is spanned by $e_1 \\wedge \\cdots \\wedge e_n$.\nFor $v_j = \\sum_i a_{ij} e_i$, $v_1 \\wedge \\cdots \\wedge v_n = \\det(a_{ij})\\, e_1 \\wedge \\cdots \\wedge e_n$.\n\n## Explanations\n\n### L1 human\n\nThe exterior algebra multiplies vectors with the wedge product, which changes sign when we swap two factors.\nIn $\\R^3$, $u \\wedge v$ describes an oriented parallelogram, and $u \\wedge v \\wedge w$ an oriented volume.\nThe top wedge product of $n$ vectors gives the determinant.\n",
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